We denote by \({\bf H}:= {\rm Hess}(\phi (y)) = \big ({\rm Hess}(u(x))\big )^{-1}\)  and notice that
\(\big ({\bf H}^{-1} + \sqrt{-1} B\big )^{-1} = {\mathbf {X}}+ \sqrt{-1} {\mathbf {Y}}\) 
with \( {\mathbf {X}}=\big ({\bf H}^{-1} + B{\bf H} B\big )^{-1}, \, \, {\mathbf {Y}}=-{\bf H} B {\mathbf {X}}\) .
Thus, we have
\(\begin{split}\delta _{\phi } \Big (\log \det \big ( {\bf H}^{-1} + \sqrt{-1} B\big )\Big )(\dot{\phi }) &= -{\rm tr}\Big (\big ({\mathbf {X}}+ \sqrt{-1} {\mathbf {Y}}\big ) {\bf H}^{-1} \dot{\bf H} {\bf H}^{-1} \Big )\\&= - {\rm tr} \Big ({\bf H}^{-1} {\mathbf {X}}{\bf H}^{-1} {\rm Hess}(\dot{\phi })\Big ) \\&=-{\rm tr} \Big (\big [{\bf H}[ {\bf H}^{-1} + B{\bf H}B ] {\bf H}\big ]^{-1}{\rm Hess}(\dot{\phi })\Big ) \\&= -{\rm tr} \Big (\big [{\bf H} + {\bf H} B{\bf H}B {\bf H}\big ]^{-1}{\rm Hess}(\dot{\phi })\Big ).\ \end{split}\) 
Notice that for any \(-invariant smooth function \) f\( on \) M\(, viewed as a function \) f(y)\( in the holomorphic coordinates \) (yj + -1 j)\(, we have\begin{equation*}{\mathrmsl d}{\mathrmsl d}^c_J f(y) = \sum _{i,j=1}^m \big ({\rm Hess}(f(y))\big )_{ij} {\mathrmsl d}y_i \wedge {\mathrmsl d}\theta _j.\end{equation*}whereas the inverse of \)\( is computed from (\ref {omega-beta}) to be\)\((\omega _{\phi })^{-1} = -\sum _{i,j=1}^m \tilde{{\mathbf {X}}}_{ij} \Big (\frac{\partial }{\partial y_i} \wedge \frac{\partial }{\partial \theta _j}\Big ) + \frac{1}{2}\tilde{{\mathbf {Y}}}_{ij} \Big (\frac{\partial }{\partial y_i} \wedge \frac{\partial }{\partial y_j} + \frac{\partial }{\partial \theta _i }\wedge \frac{\partial }{\partial \theta _j}\Big )\)\(with\)\( \tilde{{\mathbf {X}}} =\big [{\bf H} + {\bf H} B{\bf H}B {\bf H}\big ]^{-1}, \, \tilde{{\mathbf {Y}}}= B{\bf H} \tilde{{\mathbf {X}}}. \)\(We thus get\)\( \Delta ^{C}_{g_{\phi }} f = {\rm tr} \Big (\big [{\bf H} + {\bf H} B{\bf H}B {\bf H}\big ]^{-1}{\rm Hess}(f(y))\Big )\)\(and the claim follows. \)
Lemma 4.11  Let \(\phi _t\)  be a solution of (REF ). Then one has
\( \begin{split}\frac{\partial }{\partial t} \dot{\varphi }_t &= \Delta ^C_{g_t} \dot{\varphi }_t + 2\dot{\varphi }_t -2||b_t||^2_{g_t}.\end{split}\) 

Notice that by Proposition , \(\varphi _t(y) =\phi _t(y)- \phi _0(y)\)  extends to a smooth function on \(M\) , thus so is \(\dot{\phi }_t = \dot{\varphi }_t\) . For the second part, we take derivative with respect to \(t\)  in (REF ) and use Lemma REF  to compute
\(\begin{split}\frac{\partial }{\partial t} \dot{\phi }_t =&\ \Delta ^{C}_{g_t} \dot{\phi }_t + 2{\rm tr}\big [ B_t{\bf H}_t B_t{\mathbf {X}}_t\big ] + 2\dot{\phi }_t\\=&\ \Delta ^{C}_{g_t} \dot{\phi }_t + 2m -2{\rm tr}\big [{\bf H}_t^{-1}{\mathbf {X}}_t \big ]+ 2\dot{\phi }_t\\=&\ \Delta ^{C}_{g_t} \dot{\phi }_t + 2m -2{\rm tr}\big [{\rm Id} + {\bf H}_t B_t {\bf H}_t B_t\big ]^{-1} + 2\dot{\phi }_t,\end{split}\) 
where we have set \(B_t=e^{-2t}B_0\) , and \({\bf H}_t\)  and \({\mathbf {X}}_t\)  are the objects defined in the proof of Lemma REF  with respect to \(\phi _t\) .
As \(F_tJ = -g_t -b_t\) , we have
\(||F_t||^2_{g_t} = \frac{1}{2}||F_tJ||^2_{g_t} = m + ||b_t||^2_{g_t}, \) 
noting that, by our convention, the norm of a 2-form is half the norm of the corresponding tensor.
Using that \(F_t = -2{g}_t (J+ I_t)^{-1}\)  we obtain
\(2m + 2||b_t||^2_{g_t} =4||(J+ I_t)^{-1}||^2_{g_t} = -4{\rm tr}(I_t + J)^{-2}\) 
whereas by (REF ) (recalling that \(A=0\)  and \({\bf H}= {\bf \Psi }^{-1}\)  in our case) we compute
\((J+ I_t)^2\sim -4 \left( \begin{array}{cc}{\rm Id} + B_t{\bf H}_t B_t {\bf H}_t & 0 \\0 & {\rm Id} + B_t {\bf H}_t B_t {\bf H}_t\end{array} \right),\) 
so that
\(m + ||b_t||^2_{g_t} ={\rm tr}\big [{\rm Id} + B_t {\bf H}_t B_t {\bf H}_t\big ]^{-1}.\) 
The claim follows.
Using Lemma REF , we can refine the result of Lemma REF , giving a useful evolution equation for the \((2,0)\)  piece of \(b\) , which yields a key differential inequality.
Proposition 4.12  Let \((M^{2n}, g_t, I_t, J)\)  be a solution to GKRF with symplectic-type initial data, and set \(\beta _J = \pi _{\Wedge ^{2,0}_J} b\) . Then
\(\frac{\partial }{\partial t} \beta _J =&\ \Delta ^C_{g_t} \beta _J,\\\frac{\partial }{\partial t} \left| \beta _J \right|^2 =&\ \Delta ^C_{g_t} \left| \beta _J \right|^2 - \left| \nabla \beta _J \right| - \left| \overline{\partial }\beta _J \right|^2 - \left| \beta _J H^{2,1} \right|^2.\) 
We apply Lemma REF  to \(\beta _J\) . In this case, \(H= {\mathrmsl d}b\)  and \({\mathrmsl d}\beta = H^{2,1}\) , so that the term \((H\star d\beta )_{Z_1, Z_2}\)  is zero. By Lemma REF , the term \(b H\)  is exact, and thus \(({\mathrmsl d}(H \star \beta ))_{Z_1, Z_2} = (d(b H))_{Z_1, Z_2} =0\) . We conclude that
\(\Delta ^{C}_{g} \beta _J = \Big (\Delta _g \beta _J - \mathcal {L}_{\theta _J^{\sharp }} \beta _J \Big )^{2,0}.\) 
As \(\Delta _g - {\mathcal {L}}_{\theta _J^{\sharp }}\)  is a real operator, it follows from Lemma REF  that the evolution equation of \(\beta _t\)  under the GKRF is
\(\frac{\partial }{\partial t} \beta _J = \Delta ^C_{g_t} \beta _J, \) 
which is the formula obtained in [1]}, noticing that in this context \(\beta _J = \partial \alpha \)  and \(b = \Re (\partial \alpha )\)  in the notation of [1]}. The evolution of \(\left| \beta _J \right|^2\)  then follows from the established evolution for \(\beta _J\)  and a Bochner formula (cf. [3]} Lemma 4.7).

Long-time existence
In this subsection we give the proof of Theorem REF . We first derive an estimate for the tensor \(b\)  which holds without any symmetry hypotheses. Next we derive two key a priori estimates for the potential \(\phi \)  potential in the toric setting. Using these and further maximum principle arguments we prove the long time existence, giving a sharper statement yielding the nature of the estimates.
Proposition 4.13  Let \((M^{2n}, g_t, I_t, J)\)  be a solution to GKRF with symplectic-type initial data defined on \([0,T)\) . Then
\(\sup _{M \times [0,T)} \left| \beta _J \right|^2 \le \sup _{M \times \lbrace 0\rbrace } \left| \beta _J \right|^2.\) 

This follows by applying the maximum principle to the second relation in Proposition REF .
Proposition 4.14  Let \(\varphi _t\)  denote a solution to (REF ) on a toric Fano manifold. There exists a constant \(C > 0\)  so that
\(\sup _{M \times \lbrace t\rbrace } \left( \left| \varphi  \right| + \left| \dot{\varphi } \right| \right) \le C e^{2t}.\) 

Appying the maximum principle directly to (REF ) yields the upper bound \(\dot{\phi } \le C e^{2t}\) . By Proposition REF ), we have \(||b_t||^2_{g_t} \le C\) . Using this and applying the maximum principle to (REF ) yield the lower bound \(\dot{\varphi } \geqslant - C e^{2t}\) . Using the estimate of \(\dot{\varphi }\)  the bound for \(\varphi \)  follows by integration in time.
By Proposition REF  the given invariant data is equivalent to a triple \((u_0, A_0, B_0)\) . By Corollary REF  the solution to NGKRF is described by a triple \((u_t, e^{-2t} A_0, e^{-2t} B_0)\) , where \(u_t\)  is a solution to (REF ). Taking Legendre transform, by Proposition REF  this solution is equivalently described by \(\varphi _t\)  which solves (REF ). We will prove the long time existence of the flow \(\varphi _t\)  which in turn proves the long time existence of (REF ) by the discussion above.
Equation (REF ) is strictly parabolic for \(\varphi \) , thus the short-time existence is guaranteed by general theory. To establish global existence, we first aim to establish uniform parabolicity of the equation, which amounts to uniform equivalence of the time-varying metrics along the flow. We will first show an upper bound for the metric, then estimate the determinant to obtain the lower bound. To obtain the upper bound
we first recall that by , we have
\( \bigg (\frac{\partial }{\partial t} - \Delta ^C_{{g}_t}\bigg ) \log \mathop {\mathrm {tr}}\nolimits _{\overline{\omega }_0} \omega _t \le ||T^c_t||^2_{g_t} + C\mathop {\mathrm {tr}}\nolimits _{\omega _t}(\overline{\omega }_0),\) 
where \(T^c\)  is the torsion of the Chern connection. To control the two terms on the right hand side of this inequality, we will augment our test function with two auxiliary terms. We first observe that by Proposition REF , using that \(db = H\)  and dropping some negative terms we have
\( \left( \frac{\partial }{\partial t} - \Delta ^C_{g_t} \right) \left| b_t \right|^2_{g_t} \le - \left| T^c \right|^2_{g_t}.\) 
Furthermore, observe that
\( \begin{split}\bigg (\frac{\partial }{\partial t} - \Delta ^C_{g_t}\bigg )\varphi =&\ \dot{\varphi } - \mathop {\mathrm {tr}}\nolimits _{\omega _t} {\mathrmsl d}{\mathrmsl d}^c_J \varphi \\=&\ \dot{\varphi } - \mathop {\mathrm {tr}}\nolimits _{\omega _t} \overline{\omega }_t + \mathop {\mathrm {tr}}\nolimits _{\omega _t} \overline{\omega }_{0}.\end{split}\) 
Notice that
\(\mathop {\mathrm {tr}}\nolimits _{\omega _t}(\overline{\omega }_t) =\mathop {\mathrm {tr}}\nolimits \big [{\rm Id} + {\bf H}_t B_t {\bf H}_t B_t\big ]^{-1} = m + |b_t|_{g_t}^2 \le C,\) 
where the last inequality follows from Proposition REF . Using this together with Proposition REF  in (REF ) yields
\( -C + \operatorname{tr}_{\omega _t} \overline{\omega }_0 \le \left( \frac{\partial }{\partial t} - \Delta _{g_t}^C \right) \phi \le C + \operatorname{tr}_{\omega _t} \overline{\omega }_0\) 
Now set
\( \Psi _1:= \log \mathop {\mathrm {tr}}\nolimits _{\overline{\omega }_0} \omega _t + A\Big (|b_t|^2_{g_t} + \varphi \Big ). \) 
For \(A\)  chosen sufficiently large, it follows from (REF , REF , and REF ) that
\(\bigg (\frac{\partial }{\partial t} - \Delta ^C_{g_t}\bigg ) \Psi _1 \le CA.\) 
Applying the maximum principle, and using that \(\left| \phi  \right|\)  is already bounded by Proposition REF  we conclude that for any \(T > 0\)  there is a constant \(C\)  so that
\( \sup _{M \times [0,T]} \mathop {\mathrm {tr}}\nolimits _{\overline{\omega }_0} \omega _t \le C. \) 
To obtain the lower bound we get a lower bound for the volume form. First, by , we have
\( \left( \frac{\partial }{\partial t} - \Delta ^C_{g_t} \right) \log \frac{\omega _t^m}{\overline{\omega }_0^m} \geqslant - C \operatorname{tr}_{\omega _t} \overline{\omega }_0.\) 
Now set
\(\Psi _2 = \log \frac{\omega _t^m}{\overline{\omega }_0^m} + A \varphi ,\) 
By (REF ) and (REF ) we obtain
\(\left( \frac{\partial }{\partial t} - \Delta ^C_{g_t} \right) \Psi _2 \geqslant - C.\) 
Applying the maximum principle establishes the lower bound for the determinant. It follows that \(C^{-1} g_0 \le g_t \le C g_0\)  on any finite time interval. As \(|b_t|^2_{g_t}\)  is bounded, we can now apply [4]} to obtain the higher order regularity and hence the long time existence.

Monotone functionals and behavior at infinity
In this section we analyze the behavior of the normalized GKRF at infinity. First we derive an extension of Perelman's shrinker entropy to the GKRF (without toric symmetry), and use it to derive the uniform \(\kappa \) -noncollapsing estimate and convergence of nonsingular solutions of Theorem REF . Next we derive an extension of the Mabuchi \(K\) -energy to the GK setting, and show that it is monotone along the GKRF with toric symmetry. This monotonicity is the key point behind that convergence statements of Theorem REF .
Entropy monotonicity
In this subsection we observe that a modification of Perelman's entropy functional [5]} is monotone along generalized Kähler-Ricci flow for structures of symplectic type. Our discussion is a direct adaptation of [6]}, (cf. also [7]}). There the monotonicity formulae and applications are contingent upon a further monotone quantity dubbed a torsion-bounding subsolution. In the setting of generalized Kähler structures of symplectic type the torsion potential \(b\)  plays this role, and we make this explicit below
Definition 5.1  Let \((M^{2m}, g, I, J)\)  be a generalized Kähler structure of symplectic type.
Fix \(f\in C^\infty (M,{\mathbb {R}})\) , \(\tau ^*>0\) , and denote \(\tau =\tau ^*-t\) .
The
shrinker entropy associated to this data is
\(\mathcal {W}(g,H,f,\tau ) := \int _M \left[ \tau \left( \left| \nabla f \right|^2 + R -\frac{1}{12}\left| H \right|^2 \right) - \left| b \right|^2 + f - 2m \right] (4\pi \tau )^{-m}e^{-f} dV.\) 

Furthermore, let
\(\mu \left(g, H, \tau \right) := \inf _{\left\lbrace  f\ | \int _M (4 \pi \tau )^{-m}e^{-f} dV = 1\right\rbrace  } \mathcal {W}\left(g, H, f, \tau \right),\) 
and
\(\nu (g, H) := \sup _{\tau > 0} \mu _- \left(g,H,\tau \right).\) 

Proposition 5.2  Let \((M^{2m}, g_t, I_t, J)\)  be a solution to generalized Kähler-Ricci flow on a compact manifold such that \((g_0, I_0, J)\)  is symplectic type. Then, setting \(\tau = \tau ^* - t\)  for some \(\tau ^* > 0\)  we have
\(\frac{d}{d t} & \mathcal {W}(g,H,f,\tau )\\=&\ \int _M \left[ 2 \tau \left| \operatorname{Rc}- \frac{1}{4} H^2+ \nabla ^2 f - \frac{1}{2 \tau }g \right|^2 +\frac{\tau }{6} \left| d^* H + i_{\nabla f} H \right|^2 + \frac{5}{6} \left| H \right|^2\right](4 \pi \tau )^{-m} e^{-f} dV.\) 
This follows immediately by combining the results of ([6]} Proposition 6.26) noting that \(\left| b \right|^2\)  is a `torsion-bounding subsolution' via Proposition REF  and the fact that \(\left| \overline{\partial }\beta _J \right|^2 = \left| T^c \right|^2 = \left| H \right|^2\)  by Definitions REF  and REF .

Remark 5.3 Proposition REF  shows that the only self-similar solutions of the normalized GKRF with symplectic-type initial data are actually Kähler-Ricci solitons. In contrast to the shrinking case, there exist non-Kähler steady solitons for GKRF [9]}, [10]}, [11]}.

Using the entropy monotonicity of Proposition REF  the uniform \(\kappa \) -noncollapsing estimate follows from Perelman's original argument ([5]}, cf. [6]} Theorem 6.29). Now assume we have a solution to the normalized flow on \([0,\infty )\)  which satisfies a uniform curvature estimate. By the uniform \(\kappa \) -noncollapsing it follows that the volumes of unit balls are uniformly bounded below along the flow, and there is a uniform lower bound on the injectivity radius. Since the volume form is bounded above pointwise by \(F_t^m\) , the total volume is bounded above by \(\int _M F_t^m = (2 \pi c_1(M, J))^m\) . It follows that there is a uniform upper bound for the diameter along the flow. With these estimates in place, it follows from Cheeger-Gromov compactness theory (cf. [14]} Theorem 2.3 for our simplified case) that for any sequence \(\lbrace t_j\rbrace  \rightarrow \infty \)  there exists a subsequence, still denoted \(\lbrace t_j\rbrace \) , such that \((M, g_{t_j})\)  converges in Cheeger-Gromov sense to a smooth limiting Riemannian manifold \((M, g_{\infty })\) , where the limiting manifold is still \(M\)  due to the diameter upper bound. Also, it follows from ([15]}, [6]} Ch. 5) that \(H\)  and all its covariant derivatives are uniformly bounded along the flow, and thus by choosing a further subsequence \(H\)  can be assumed to converge to \(H_{\infty }\) . We claim that by choosing a further subsequence the generalized Kähler structure also converges, which requires establishing \(C^{\infty }\)  estimates for \(I\)  and \(J\) . First, the \(g\) -norms of \(I\)  and \(J\)  are uniformly bounded since \(g\)  is compatible with both. By a standard argument using the fact that the Bismut \(\nabla ^{B,I}\)  preserves \(I\) , and that the coefficients of this connection are already converging by the discussion above, it follows that there are uniform \(C^{\infty }\)  estimates for \(I\) , and the same argument using \(\nabla ^{B,J}\)  yields estimates for \(J\) . It follows that we can choose a further subsequence such that \((M, g_{t_j}, I_{t_j}, J)\)  converges in Cheeger-Gromov sense to a limiting generalized Kähler manifold \((M, g_{\infty }, I_{\infty }, J_{\infty })\) .
Now note that by rescaling the result of Proposition REF  holds for the normalized flow, setting \(\tau \equiv 1\) . This also implies that the corresponding functional \(\mu \)  is monotone increasing. Since the solution is nonsingular, there are uniform bounds on \(\mu \)  and all of its derivatives. It thus follows that for an arbitrary sequences of times \(\lbrace t_j\rbrace  \rightarrow \infty \)  one has
\(\lim _{j \rightarrow \infty } \left| \frac{d}{dt} \mu (g_{t_j}, H_{t_j}) \right| = 0.\) 

For such a sequence \(\lbrace t_j\rbrace \) , let \((M, g_{\infty }, I_{\infty }, J_{\infty })\)  denote the generalized Kähler limit constructed as above. It follows from the entropy monotonicity formula that the limiting torsion \(H_{\infty }\)  must vanish, and furthermore \(g_{\infty }\)  is a shrinking soliton. Since \(H_{\infty }\)  vanishes, the pairs \((g_{\infty }, I_{\infty })\)  and \((g_{\infty }, J_{\infty })\)  are both Kähler; this shows that the symplectic 2-form \(F_{\infty }\)  taming \(I_{\infty }\)  and \(J_{\infty }\)  is parallel with respect to \(g_{\infty }\) , and hence must be a \((1, 1)\) -form by the Fano condition of \((M, J_{\infty })\) . It then follows that we must have \(I_{\infty } = J_{\infty }\) .

The extended Mabuchi functional
In this section we give an extension of the Mabuchi functional to the setting of GK structures of symplectic type on a Kähler background. To begin we first observe that it is possible to canonically associate a Kähler metric to every symplectic-type GK structure on a Kähler manifold. First, using Hodge decomposition, the deRham class \(\alpha = [F]\)  defines an element \(\overline{\alpha }\in H^{1,1}(M, {\mathbb {R}})\) . Using that \(F\)  tames \(J\)  and a result of Demailly-Paun [17]}, it follows that \(\overline{\alpha }\)  is a Kähler class of \((M, J)\) . By Yau's theorem [18]}, we can find a unique Kähler metric \(\overline{\omega }\in \overline{\alpha }\)  with the property
\(\overline{\omega }^{[m]} = c F^{[m]}, \qquad c =\frac{\overline{\alpha }^{[m]}}{[F]^{[m]}}.\) 
Definition 5.4  On a Kähler background \((M, J)\) , for a GK structure \((g, I, J)\)  we shall refer to the unique Kähler metric \(\overline{\omega }\)  satisfying (REF ) as the Kähler reduction of \((g, I, J)\) . Fixing a background Kähler metric \(\overline{\omega }_0\) , we refer to the Kähler reduction of a given \(F\)  as \(\overline{\omega }=\overline{\omega }_{\varphi }= \overline{\omega }_0+ {\mathrmsl d}{\mathrmsl d}^c_J \varphi \) , where we recall, the smooth function \(\varphi \)  is defined only up to an additive constant.

Remark 5.5 Kähler reductions exist in smooth families along a solution to GKRF of symplectic type. In general it might not be possible to derive an explicit formula for the evolution of \(\overline{\omega }\) , but in the toric Fano case this is what is achieved in Proposition REF .

We now recall the definitions of the Aubin-Mabuchi functionals \(I\)  and \(I^{\rho _0}\)  which are defined on the space of \(\overline{\omega }_0\) -relative Kähler potentials \(\varphi \)  by
\(\begin{split}(\delta _{\varphi } I) (\dot{\varphi }) &=\int _M \dot{\varphi }\overline{\omega }_{\varphi }^{[m]}, \hspace{34.14322pt} \qquad I(0)=0, \\(\delta _{\varphi } I^{\rho _0} ) (\dot{\varphi }) &=\int _M \dot{\varphi }\rho _0 \wedge \overline{\omega }_{\varphi }^{[m-1]}, \qquad I^{\rho _0}(0)=0\end{split} \) 
where \(\rho _0\)  is the Ricci form of \(\overline{\omega }_0\) . As \(\rho _0 \in 2\pi c_1(M, J)\) ,
letting \(a:= 4\pi \frac{c_1(M, J) \cdot [\overline{\alpha }]^{[m-1]}}{[\overline{\alpha }]^{[m]}}\)  the expression
\(a I(\varphi ) - 2I^{\rho _0}(\varphi )\)  does not change if we add to \(\varphi \)  a real constant, i.e. it introduces a functional, denoted by \(\left(aI-2I^{\rho _0}\right)(\overline{\omega }_{\varphi })\) , acting on the Kähler metrics \(\overline{\omega }_{\varphi } \in \overline{\alpha }\) .
Definition 5.6  For \(F\in {\bf GK}(M, J, \alpha )\)  we define the extended Mabuchi energy by
\(\begin{split}{\bf M}(F) &:= \left(\frac{\overline{\alpha }^{m}}{[F]^{m}}\right)\int _M \left[ \log \left(\frac{F^{[m]}}{\overline{\omega }_0^{[m]}}\right) + \log \left(\frac{F^{[m]}}{\omega _J^{[m]}}\right)\right]F^{[m]} + \left(a I- 2I^{\rho _0}\right)(\overline{\omega }_{\varphi }), \\a &:= 4\pi \frac{c_1(M, J) \cdot \overline{\alpha }^{[m-1]}}{\overline{\alpha }^{[m]}},\end{split}\) 

Remark 5.7  In the case when \(\alpha = \overline{\alpha }\)  is a Kähler class, one has that \(\overline{\omega }_{\varphi }^{[m]}=F^{[m]} \ge \omega _J^{[m]}\) . This implies the basic inequality
\({\bf M}(F) \ge {\bf M}(\overline{\omega }_{\varphi }),\) 
where \({\bf M}(\overline{\omega }_{\varphi })\)  is the usual Mabuchi energy of the Kähler metric \(\overline{\omega }_{\varphi } \in \alpha \) , see e.g. [19]}.


The extended Mabuchi energy in the toric case
On a compact smooth toric Kähler manifold \((M, J, \omega _0, \)  with Delzant polytope \(({\mathrm {P}}, {{\mathrm {L}}})\) ,
Donaldson [20]} found concise expressions for the Futaki invariant and the Mabuchi energy in terms of \(({\mathrm {P}}, {\mathrm {L}})\)  and the space \({\mathcal {S}}({\mathrm {P}}, {\mathrm {L}})\) . We use this point of view in order to express the extended Mabuchi energy as a functional acting on the data \((u, A, B)\)  associated to \(({\mathrm {P}}, {\mathrm {L}})\) . Following [20]}, we introduce a linear functional defined on the space of continuous functions on \({\mathrm {P}}\)  by
\({\bf F} (f) := - a\int _{{\mathrm {P}}} f(x){\mathrmsl d}x + 2 \int _{\partial {\mathrm {P}}} f(x) {\mathrmsl d}\sigma _{{\mathrm {L}}}, \qquad a:= 2\frac{\operatorname{Vol}\left({\partial {\mathrm {P}}}, {\mathrmsl d}\sigma _{{\mathrm {L}}}\right)}{\operatorname{Vol}({\mathrm {P}}, {\mathrmsl d}x)},\) 
where, in the above expression, \({\mathrmsl d}x\)  is a fixed Lebesgue measure on \({\mathfrak {t}}^*\)  (associated with a chosen basis of \({\mathfrak {t}}\) ) and \({\mathrmsl d}\sigma _{{\mathrm {L}}}\)  is the induced measure by \({\mathrmsl d}x\)  and the inward normal \({\mathrmsl d}L_j\)  on each facet \({\mathrm {P}}_j \subset \partial {\mathrm {P}}\) . If we take \(f(x)= \langle \xi , x \rangle + \lambda \)  to be affine-linear, (REF ) computes, up to a dimensional multiplicative constant, the Futaki invariant of the holomorphic vector \(JK_{\xi }\)  on \((M, J, [\omega _0])\) .
Lemma 5.8  Let \((g, I, J)\)  be a toric generalized Kähler structure, corresponding to the data \((u, A, B)\)  with respect to \(\omega _0\) . Then
\(\frac{1}{(2\pi )^m}{\bf M}(F)= {\bf F}(u) - \int _{{\mathrm {P}}} \log \det \Big ({\rm Hess}(u) + \sqrt{-1}B\big ) {\mathrmsl d}x + \int _{{\mathrm {P}}} \log \det \Big ({\rm Hess}(u_0)\big ) {\mathrmsl d}x, \) 
where \(u_0\)  is the symplectic potential of a background toric Kähler metric.

On a toric manifold \(H^2(M, =H^{1,1}(M, \) , so we are in the case \(\alpha =\overline{\alpha }\)  with \(F^{[m]}=\overline{\omega }_{\varphi }^{[m]}\) . By (REF ) and using (REF ), we compute
\(\begin{split}{\bf M}(F) &= {\bf M}(\overline{\omega }_{\varphi }) +(2\pi )^m \int _{{\mathrm {P}}} \log \left(\frac{\det \mathrm {Hess}(u)}{\det (\mathrm {Hess}(u) + \sqrt{-1}B)}\right) {\mathrmsl d}x.\end{split}\) 
The claim follows by the above, taking in mind that \(\overline{\omega }_{\varphi } = {\mathrmsl d}{\mathrmsl d}^c_{J_A} \phi (\tilde{y})\) , where \(\phi \)  is the Legendre transform of \(u\in {\mathcal {S}}({\mathrm {P}}, {\mathrm {L}})\)  (see ()) and \(\tilde{y}\)  are the pluriharmonic coordinates of \(J_A\)  (see (REF )), and the fact that (see [20]})
\({\bf M}(\overline{\omega }_{\varphi }) = (2\pi )^m\left({\bf F}(u) -\int _{{\mathrm {P}}} \log \det (\mathrm {Hess}(u)) {\mathrmsl d}x + \int _{{\mathrm {P}}} \log \det (\mathrm {Hess}(u_0)){\mathrmsl d}x \right) .\) 
We next derive fundamental variational properties for the extended Mabuchi energy. Recall our expression for the metric
\({\bf X} = \Re \big [{\rm Hess}(u) + \sqrt{-1}B\big ]^{-1}.\) 

It turns out that the quantity
\(\kappa (u, A, B):= -\sum _{i.j=1}^m {\bf X}_{ij,ij}\) 
is identified in [23]}, [24]} with the momentum map for the action of the group of \(-equivariant Hamiltonian symplectomorphisms of \) (M, 0)\( on the space of compatible toric generalized Kähler structures, and thus \)\( extends the notion of scalar curvature to the generalized Kähler context, via the momentum map picture of \cite {Donaldson, fujiki}. We call \)\( the \emph {generalized scalar curvature}. This point of view is extended to generalized Kähler manifolds of symplectic type (without toric symmetry) in \cite {Goto-GIT}. We thus conclude that the critical points of \)M\( correspond to toric GK structures for which \) Gscal(g)=(u, A, B)=a\(.\)
Proposition 5.9  Given a one-parameter family of symplectic potentials \(u_s = u + s \dot{u}\)  and fixed matrices \(A,B\) , we obtain
\( (\delta _{u} {\bf M})(\dot{u}) = \int _{{\mathrm {P}}} \dot{u} \left(\kappa (u,A,B) - a\right) {\mathrmsl d}x.\) 
In particular, the critical points of \({\bf M}\)  on the space of toric generalized Kähler structures of symplectic type on \((M, J)\)  with fixed cohomology class and holomorphic Poisson tensor have constant generalized Kähler scalar curvature. Furthermore,
\((\delta ^2_{u} {\bf M})(\dot{u}, \dot{u}) = \int _{{\mathrm {P}}} {\rm tr}\Big [\Big (\big ({\rm Hess}(u) + \sqrt{-1}B\big )^{-1}{\rm Hess}(\dot{u})\Big )^2\Big ] {\mathrmsl d}x.\) 

These follow by observing that the linearization of the entropy term is
\(\int _{{\mathrm {P}}} {\rm tr} \Big [ \big ({\rm Hess}(u) + \sqrt{-1}B\big )^{-1} \circ {\rm Hess}(\dot{u})\Big ] {\mathrmsl d}x = \int _{{\mathrm {P}}} {\rm tr}\big ({\bf X} \circ {\rm Hess}(\dot{u})\big ) {\mathrmsl d}x.\) 
Using that \({\bf F}\)  is linear, and differentiating one more time the above expression yields the formula for the Hessian of \({\bf M}\) . To obtain the first formula, we need to integrate by parts the RHS of the above expression as in [20]}. To this end, we observe that \({\bf X}= \big [{\rm Hess}(u) + B \big ({\rm Hess}(u)\big )^{-1}B\big ]^{-1}\)  is a smooth symmetric matrix on \({\mathrm {P}}\) , which satisfies the first order boundary conditions of  (see also Remark 4 in that reference). The second variation follows easily by passing another \(s\)  derivative through the variation formula above and using that \(\ddot{u} = 0\) .
Remark 5.10  By the convexity property of \({\bf M}\)  established in Lemma REF  (and using that \({\mathcal {S}}({\mathrm {P}}, {\mathrm {L}})\)  is linearly convex) for given \(A,B\) , there exists at most one, up to an addition with an affine-linear function, \(u\in {\mathcal {S}}({\mathrm {P}}, {{\mathrm {L}}})\)  which satisfies \(\kappa (u, A, B)=a\) . One can also show that for any \(u\in {\mathcal {S}}({\mathrm {P}}, {{\mathrm {L}}})\) , the corresponding \({\bf X}\)  is a smooth, matrix-valued function on \({\mathrm {P}}\)  which satisfies the positivity and boundary conditions of , and thus (REF ) gives rise to an almost Kähler structure with constant scalar curvature in the sense of [26]}. The necessary condition for the existence of such \(u\)  is the one of uniform relative K-stability of \(({\mathrm {P}}, {{\mathrm {L}}})\)  (see [20]}, [28]}, [29]}) which implies that there exist constants \(\lambda , \delta >0\)  such that
\({\bf F}(u) \ge \lambda \int _{{\mathrm {P}}} |\mathring{u}| {\mathrmsl d}x -\delta , \qquad \forall u \in {\mathcal {S}}({\mathrm {P}}, {{\mathrm {L}}}),\) 
where \(\mathring{u}(x) = u(x) + \ell (x)\)  for an affine-linear function \(\ell (x)\)  is the unique normalization of \(u\) , i.e. which satisfies \(\mathring{u}(x) \ge \mathring{u}(x_0)=0\)  for a fixed point \(x_0\)  in the interior of \({\mathrm {P}}\) .
Conversely, it is now established (as a consequence of deep recent work by Chen-Cheng [30]} and previous work by Zhou-Zhu [29]}) that any smooth toric variety with uniform relative K-stable Delzant polytope \(({\mathrm {P}}, {{\mathrm {L}}})\)  in the sense of (REF ) admits a compatible toric Kähler metric of constant scalar curvatue, thus showing that the existence of toric generalized Kähler structures of constant generalized Kähler scalar curvature is equivalent to the existence of Kähler ones.

Lastly in this subsection we establish the monotonicity of the extended Mabuchi energy along NGKRF in the toric Fano setting. We build this up through a series of lemmas.
Lemma 5.11  On a toric Fano manifold \((M, J, \) , along the flow (REF )
\( \frac{d}{dt}{\bf M}(F_{\varphi _t}) = \left[-\int _M |d \dot{\varphi }_t|^2_{\overline{\omega }_{\varphi _t}} F_{\varphi _t}^{[m]} + \int _M \left(s_J^B(g_{\varphi _t}) - 2m\right)F_{\varphi _t}^{[m]}\right],\) 
where \(s_J^B(g_{\varphi _t})\)  denotes the Bismut scalar curvature of the pluriclosed Hermitian structure \((g_{\varphi _t}, J)\) .

To simplify the notation, we let \(\overline{\omega }_t=\overline{\omega }_{\varphi _t}\) , \(F_t =F_{\varphi _t}\)  and \(\omega _t=\omega _{\varphi _t}\) . Referring to (REF ) (with \(\alpha =\overline{\alpha }\)  and \(a=2m\)  according to our normalization), \(F_t^m =\overline{\omega }_t^m\)  and using (REF )-(REF ) (with \(c(t) \equiv 0\) ), we compute
\(\begin{split}\frac{d}{dt} {\bf M}(F_t) =&\ 2 \int _M \operatorname{tr}_{\omega _t}\left(\rho ^B_J(g_t) - \overline{\omega }_t\right)\overline{\omega }_t^{[m]} +\int _M\big (\dot{\varphi }_t - 2\varphi _t + 2h_0\big ) ({\mathrmsl d}{\mathrmsl d}^c\dot{\varphi }_t) \wedge \overline{\omega }_t^{[m-1]} \\&\ \qquad + 2m \int _M \dot{\varphi }_t \overline{\omega }_t^{[m]} - 2\int _M \dot{\varphi }_t \rho _0 \wedge \overline{\omega }_t^{[m-1]}\\=&\ \int _M\left(s_J^B(g_t)- 2m\right)\overline{\omega }_t^{[m]} -\int _M|d\dot{\varphi }_t|^2_{\overline{\omega }_t}\overline{\omega }_t^{[m]} \\&\ \qquad -2\int _M \dot{\varphi }_t(\overline{\omega }_t - \overline{\omega }_0)\wedge \overline{\omega }_t^{[m-1]} + 2\int _M\dot{\varphi }_t(\rho _0-\overline{\omega }_0)\wedge \overline{\omega }_t^{[m-1]} \\&\ \qquad +2m \int _M \dot{\varphi }_t \overline{\omega }_t^{[m]} - 2\int _M \dot{\varphi }_t \rho _0 \wedge \overline{\omega }_t^{[m-1]} \\=&\ \int _M\left(s_J^B(g_t)- 2m\right)\overline{\omega }_t^{[m]} -\int _M|d\dot{\varphi }_t|^2_{\overline{\omega }_t}\overline{\omega }_t^{[m]}.\end{split}\) 
The claim follows by using \(\overline{\omega }_t^{[m]} =F_t^{[m]}\)  again.
Lemma 5.12  Suppose \((M, g, I, J)\)  is a compact generalized Kähler manifold of symplectic type. Then
\(\int _M\left[\operatorname{tr}_{\omega _J}(\rho ^B_J) - \operatorname{tr}_{F}(\rho ^B_J)\right]F^{[m]} = -\frac{1}{8}\int _M\left[ \Big |{\mathrmsl d}\log \det (I+J)-\theta _I-\theta _J\Big |^2_g+|\theta _I-\theta _J|^2_g \right]F^{[m]}.\) 

We denote by \(s_J^B\)  and \(s_I^B\)  the Bismut-Ricci scalar curvatures of \((g, J)\)  and \((g, I)\) , respectively. By Proposition REF  and [32]}, we know that
\( \rho ^B_J- \rho ^B_I = \frac{1}{2}{\mathcal {L}}_{(\theta _J^{\sharp }-\theta _{I}^{\sharp })} F, \qquad s_J^B - s_I^B = -\delta ^g(\theta _J - \theta _I). \) 
Using that \(F=-2g(I+J)^{-1}\)  and the above formula, we compute
\(\begin{split}2\left[\operatorname{tr}_{\omega _J}(\rho ^B_J) - \operatorname{tr}_{F}(\rho ^B_J)\right] &= s_J^B - \big \langle \rho ^B_J, \omega _J + \omega _I\big \rangle _g = \frac{1}{2} s_J^B - \big \langle \rho ^B_J, \omega _I\big \rangle _g \\&= \frac{1}{2} (s_J^B - s_I^B) - \frac{1}{2}\Big \langle {\mathcal {L}}_{(\theta _J^{\sharp }-\theta _{I}^{\sharp })} F, \omega _I\Big \rangle _g\\=&\ -\frac{1}{2}\delta ^g(\theta _J - \theta _I)- \frac{1}{2}\Big \langle {\mathcal {L}}_{(\theta _J^{\sharp }-\theta _{I}^{\sharp })} F, \omega _I\Big \rangle _g.\end{split} \) 
We next use that, by (REF ),
\({\mathcal {L}}_{(\theta _J^{\sharp }-\theta _{I}^{\sharp })} F = {\mathrmsl d}\left(F(\theta _J^{\sharp }-\theta _I^{\sharp })\right) = {\mathrmsl d}\left(I(\theta _J - \theta _I) + Ib(\theta _J^{\sharp }-\theta _I^{\sharp })\right),\) 
so we continue the computation
\(\begin{split}2\left[\operatorname{tr}_{\omega _J}(\rho ^B_J) - \operatorname{tr}_{F}(\rho ^B_J)\right] &= -\frac{1}{2}\delta ^g(\theta _J - \theta _I)- \frac{1}{2}\Big \langle {\mathrmsl d}\left(I\left( (\theta _J - \theta _I) + b(\theta _J^{\sharp }-\theta _I^{\sharp })\right)\right), \omega _I\Big \rangle _g \\&= \frac{1}{2}\left(\delta ^g\left(b(\theta _J^{\sharp }-\theta _I^{\sharp })\right) + \big \langle \theta _J - \theta _I + b(\theta _J^{\sharp }-\theta _I^{\sharp }), \theta _I\big \rangle _g\right),\end{split}\) 
where we have used the identity
\(\langle {\mathrmsl d}I \alpha , \omega _I\rangle _g= -\delta ^g \alpha - \langle \alpha , \theta _I\rangle _g.\) 
We next use the identity (which follows from Lemma REF )
\( {\mathrmsl d}\log \det (I+J) = \theta _I + \theta _J + b(\theta _J^{\sharp }-\theta _I^{\sharp })\) 
in order to express the terms containing \(b(\theta _J^{\sharp }-\theta _I^{\sharp })\) . We thus get
\(\begin{split}2\left[\operatorname{tr}_{\omega _J}(\rho ^B_J) - \operatorname{tr}_{F}(\rho ^B_J)\right] &= \frac{1}{2}\left(-\Delta ^g \log \det (I+J) - \delta ^g(\theta _I +\theta _J) + \big \langle - 2\theta _I + {\mathrmsl d}\log \det (I+J), \theta _I\big \rangle _g\right) \\&= \frac{1}{2}\left(-\Delta ^g \log \det (I+J) - \delta ^g(\theta _I +\theta _J) - 2|\theta _I|^2_g + \big \langle {\mathrmsl d}\log \det (I+J), \theta _I\big \rangle _g\right).\end{split}\) 
We finally use that (as \(F= -2g(I+J)^{-1}\) )
\(F^{[m]} = 2^m \det (I+J)^{-\frac{1}{2}}\, \omega _J^{[m]} \) 
and integration by parts to get
\(\begin{split}&\int _M\left[\operatorname{tr}_{\omega _J}(\rho ^B_J) - \operatorname{tr}_{F}(\rho ^B_J)\right] F^{[m]} \\&= 2^{m-2}\int _M\left(-\Delta ^g \log \det (I+J) + \big \langle {\mathrmsl d}\log \det (I+J), \theta _I\big \rangle _g - \delta ^g(\theta _I +\theta _J) - 2|\theta _I|^2_g \right)\det (I+J)^{-\frac{1}{2}}\, \omega _J^{[m]} \\&= 2^{m-2}\int _M\left(-\Delta ^g \log \det (I+J) + \big \langle {\mathrmsl d}\log \det (I+J), \theta _I\big \rangle _g - |\theta _I|^2_g-|\theta _J|^2_g-2\delta ^g\theta _J \right)\det (I+J)^{-\frac{1}{2}}\, \omega _J^{[m]} \\&= 2^{m-2}\int _M\left[-\frac{1}{2}\Big |{\mathrmsl d}\log \det (I+J)\Big |^2_g + \big \langle {\mathrmsl d}\log \det (I+J), \theta _I+\theta _J\big \rangle _g-|\theta _I|^2_g-|\theta _J|^2_g \right]\det (I+J)^{-\frac{1}{2}}\, \omega _J^{[m]} \\&= -\frac{1}{8}\int _M\left[ \Big |{\mathrmsl d}\log \det (I+J)-\theta _I-\theta _J\Big |^2_g+|\theta _I-\theta _J|^2_g \right]F^{[m]}\end{split}\) 
where we have used the relation \(\delta ^g \theta _I + |\theta _I|^2_g =|H|_g^2=\delta ^g\theta _J+|\theta _J|^2_g\)  from [32]}.
Corollary 5.13  On a toric Fano manifold, the Mabuchi functional \({\bf M}(F_{\varphi _t})\)  is monotone decreasing along NGKRF.

This follows directly from Lemma REF  and Lemma REF , noting that \(\operatorname{tr}_{\omega }(\rho ^B_J(g))= \frac{s_J^B}{2}\)  and
\(\int _M \operatorname{tr}_{F}(\rho _J^B(g)) F^{[m]} =\int _M \rho ^B_J(g)\wedge F^{[m-1]} = m\) 
by the Fano condition and the normalization \([F]\in 2\pi c_1(M, J)\) .

Weak convergence
We suppose through this section that \((M, J, \)  is a smooth toric Fano manifold, \(\overline{\omega }_0 \in 2\pi c_1(M, J)\)  is an initial Kähler form with corresponding barycentered Fano polytope \(({\mathrm {P}}, {{\mathrm {L}}})\) , and we further assume that the Futaki invariant of \((M, J, [\overline{\omega }_0])\)  vanishes. In this case, by [34]}, \((M, J)\)  admits a Kähler-Einstein metric and, by [35]}, [36]}, the Mabuchi energy is coercive relative to the connected component of the identity \(\mathrmsl {Aut}_{0}(M, J)\)  of the group of automorphisms of \((M, J)\) . We first recall what this precisely means. Denote by
\( {\mathcal {H}}(M, \overline{\omega }_0)=\left\lbrace  \varphi \in C^{\infty }(M)\, | \, \overline{\omega }_{\varphi }:= \overline{\omega }_0 + {\mathrmsl d}{\mathrmsl d}^c \varphi >0 \right\rbrace \) 
the space of \(\overline{\omega }_0\) -relative Kähler potentials on \(M\) , and consider the Aubin-Mabuchi functional \(I\)  on \({\mathcal {H}}(M, \overline{\omega }_0)\) , defined in (REF ). Notice that \(I(\varphi + c) = I(\varphi ) + c\operatorname{Vol}(M, \overline{\omega }_0)\) , so we can introduce the slice
\(\mathring{{\mathcal {H}}}(M, \overline{\omega }_0) := {\mathcal {H}}(M, \overline{\omega }_0) \cap I^{-1}(0),\) 
and consider the induced action of \({\mathbb {G}}:=\mathrmsl {Aut}_0(M, J)\)  on \(\mathring{{\mathcal {H}}}(M, \overline{\omega }_0)\)  via the natural action of \({\mathbb {G}}\)  on the space of \(-invariant Kähler metrics in \) [0]\( by pullbacks, i.e. for any \) H(M, 0)\( and any \) G\(, we let \) []\( denote the uniquely determined \)0\(-relative Kähler potential in \) H(M, 0)\( of the Kähler metric \) *()\(. We thus have\)\(\tau [\varphi ] = \tau [0] + \varphi \circ \tau . \)\(\) We next consider the \(d_1\) -distance on \({\mathcal {H}}(M, \overline{\omega }_0)\) , introduced via the \(L_1\) -length of a smooth curve \(\varphi (t )\in {\mathcal {H}}(M, \overline{\omega }_0)\) :
\(l_1(\varphi (t)) = \int _{0}^1 \int _M |{\dot{\varphi }}(t)| \, (\overline{\omega }_{\varphi (t)})^{[m]}.\) 
By the results in [37]},
\(d_1(\varphi _0, \varphi _1) := \inf \Big \lbrace l_1(\varphi (t))\, \Big | \, \varphi (t, x) \in C^{\infty }([0, 1] \times M), \, \varphi (t, \cdot ) \in {\mathcal {H}}(M, \overline{\omega }_0), \, \varphi (0)=\varphi _0, \, \varphi (1)=\varphi _1 \Big \rbrace \) 
defines a distance.
We then have
Theorem 5.14 [35]} If \((M, J, [\overline{\omega }_0])\)  admits a Kähler-Einstein metric, then the Mabuchi energy \({\bf M}\)  is \({\mathbb {G}}\) -invariant and \({\mathbb {G}}\) -coercive on \({\mathcal {H}}(M, \overline{\omega }_0)\) , i.e. there are uniform constants \(\lambda >0, \delta \)  such that
\( {\bf M}(\varphi ) \ge \lambda \inf _{\tau \in {\mathbb {G}}} d_1(0, \tau [\varphi ]) - \delta , \qquad \forall \varphi \in \mathring{{\mathcal {H}}}(M, \overline{\omega }_0). \) 

An important ingredient in proving the above result is the weak compactness, established in , [39]}. To state it, we denote by
\(\mathrm {PSH}(M, \overline{\omega }_0)\)  the space of \(\overline{\omega }_0\) -relative plurisubharmonic functions on \(M\) , i.e. the usc functions \(\varphi \in L^1(M, \overline{\omega }_0)\)  such that \(\overline{\omega }_{\varphi }=\overline{\omega }_0 + {\mathrmsl d}{\mathrmsl d}^c \varphi \ge 0\)  in the sense of currents. Furthermore, we denote by \({\mathcal {E}}(M, \overline{\omega }_0) \subset \mathrm {PSH}(M, \overline{\omega }_0)\)  the subspace of elements of full Monge-Ampère mass, i.e.
\( {\mathcal {E}}(M, \overline{\omega }_0) = \Big \lbrace  \varphi \in \mathrm {PSH}(M, \overline{\omega }_0) \, \Big | \, \int _M \overline{\omega }_{\varphi }^{[m]} = \int _M \overline{\omega }_0^{[m]}=\operatorname{Vol}(M, \overline{\omega }_0)\Big \rbrace .\) 
By the results in [40]}, [41]}, each \(\varphi \in {\mathcal {E}}(M, \overline{\omega }_0)\)  can be weakly approximated (as a current) by a sequence of smooth functions \((\varphi _j)_j \in {\mathcal {H}}(M, \overline{\omega }_0)\)  and for any such sequence \(\lim _{j\rightarrow \infty } \overline{\omega }_{\varphi _j}^{[m]} = \overline{\omega }_{\varphi }^{[m]}\)  in the sense of measures. We can take the latter as a definition of \(\overline{\omega }_{\varphi }^{[m]}\)  when \(\varphi \in {\mathcal {E}}(M, \overline{\omega }_0)\) . Following [40]}, one further introduces the sub-space of elements with full Monge-Ampère mass and finite energy
\({\mathcal {E}}^1(M, \overline{\omega }_0) = \Big \lbrace  \varphi \in {\mathcal {E}}(M, \overline{\omega }_0) \, \Big | \, \int _M |\varphi | \, \overline{\omega }_{\varphi }^{[m]} < \infty \Big \rbrace .\) 
The central fact [37]} in this theory is that \(d_1\)  extends to define a complete distance on \({\mathcal {E}}^1(M, \overline{\omega }_0)\) , such that \(({\mathcal {E}}^1(M, \overline{\omega }_0), d_1)\)  is a complete geodesic space in which \({\mathcal {H}}(M, \overline{\omega }_0)\)  is densely embedded, and the \(d_1\) -convergence on \({\mathcal {E}}^1(M, \overline{\omega }_0), d_1)\)  is stronger than both the \(L^1(M, \overline{\omega }_0)\)  and the weak convergence of \((1,1)\) -currents. Furthermore, \({\bf M}\)  naturally extends to a continuous and lsc functional on \({\mathcal {E}}^1(M, \overline{\omega }_0)\) , such that
Theorem 5.15 , [39]} If \((\varphi _j)_j \in {\mathcal {E}}^1(M, \overline{\omega }_0)\)  is a sequence with
\( d_1(0, \varphi _j) <C, \qquad {\bf M}(\varphi _j) < C\) 
then there exists a subsequence \((\varphi _{j_k})_k\)  which converges with respect to \(d_1\)  to an element \(\varphi _0 \in {\mathcal {E}}^1(M, \overline{\omega }_0)\) .

In the toric case considered in this paper, we shall rather work with the subspace
\({\mathcal {H}}_{(M, \overline{\omega }_0) := {\mathcal {H}}(M, \overline{\omega }_0) \cap C^{\infty }_{(M) of -invariant relative Kähler potentials, and consider instead of {\mathbb {G}}= \mathrmsl {Aut}_0(M, J) the complex torus {. The coercivity principle of \cite [Theorem~3.4]{DR} still applies in the -relative setting, noting that \mathrmsl {Aut}_0(M, J) is maximal and  acts transitively on the space of the -invariant Kähler-Einstein metrics in [\overline{\omega }_0] (see \cite {guan} for the toric case and \cite [Thm.~2]{lahdili} for a more general setting), so we have the following -relative version of Theorem~\ref {DR} (see \cite {hisamoto,ZZ} or \cite [Prop.~3.4]{apostolov-notes} for the toric case, and \cite [Thm.~1.6]{HL} for a more general statement):}\begin{thm} If (M, J, [\overline{\omega }_0],  admits a Kähler-Einstein metric, then the Mabuchi energy {\bf M}: {\mathcal {H}}_{(M, \overline{\omega }_0) \rightarrow {\mathbb {R}} is {-invariant and {-coercive, i.e. there are uniform constants \lambda >0, \delta  such that {\bf M}(\varphi ) \ge \lambda \inf _{\tau \in {} d_1(0, \tau [\varphi ]) - \delta , \qquad \forall \varphi \in \mathring{{\mathcal {H}}}_{(M, \overline{\omega }_0). }}We next use Theorem~\ref {DT-toric} in order to get a weak convergence of the global solution of NGKRF in the toric Kähler-Einstein case.}{\begin{xmlelement*}{proof}Let \varphi _t := \phi _t - \phi _0 \in {\mathcal {H}}_{(M, \omega _0) be the relative Kähler potentials along the Kähler reduction (\ref {unnormalized}) of the normalized generalized Kähler Ricci flow, and \mathring{\varphi }_t \in \mathring{{\mathcal {H}}}_{(M, \overline{\omega }_0) ={\bf I}^{-1}(0) the corresponding normalized \overline{\omega }_0-relative Kähler potentials. Using Theorem~\ref {DT-toric}, Corollary~\ref {Mabuchi-monotone} and (\ref {M-tames}), we have along the Kähler reduction (\ref {unnormalized}) of the normalized generalized Kähler Ricci flow, \lambda \inf _{\tau \in {} d_1(0, \tau [\mathring{\varphi }_t]) \le {\bf M}(\overline{\omega }_{\varphi _t}) + \delta \le {\bf M}(F_{\varphi _t}) + \delta \le {{\bf M}}(F_0)+ \delta .Furthermore, by \cite [Prop.~6.8]{DR} (using that { is reductive), for each t\in [0, \infty ) there exists \tau _t \in { such that\inf _{\tau \in {} d_1(0, \tau [\mathring{\varphi }_t])= d_1(0, \tau _t[\mathring{\varphi }_t]). We thus get uniform estimates d_1(0, \tau _t[\mathring{\varphi }_t]) \le C, \qquad {\bf M}(\tau _t[\mathring{\varphi }_t])= {\bf M}(\varphi _t) < C which, by Theorem~\ref {compactness} and the fact that {\mathcal {E}}^1_{(M, \overline{\omega }_0) is d_1-closed as a subset of {\mathcal {E}}^1(M, \overline{\omega }_0)~\cite {DR}, proves the first part of the theorem.}For the second part, according to \cite {BS}, on a symmetric toric Fano manifolds the Tian \alpha -invariant \cite {Tian} satisfies \alpha _{(M, J)>\frac{m}{m+1}. However, the above estimate of \alpha _{(M, J) also yields (see e.g. \cite [Theorem 1.4]{Dervan}) that the Mabuchi energy is \emph {strongly} coercive, i.e. that there exist constants \lambda >0, \delta  such that {\bf M}(\overline{\omega }_{\varphi }) \ge \lambda d_1(0, \mathring{\varphi }) - \delta , \qquad \forall \varphi \in {\mathcal {H}}_{(M, \overline{\omega }_0), where we have used \cite [Prop. 5.4]{DR} to connect with the definition of coercivity in terms of the J-functional used in \cite {Dervan}. The same argument as above concludes the proof.}}}\begin{rmk} One would expect to further improve the unconditional weak convergence in Theorem~\ref {t:convthm}, by showing that the limiting curent \overline{\omega }_{\infty } is in fact a Kähler--Einstein metric. In the case of the normalized KRF on a Fano manifold with strongly coercive Mabuchi energy, such a conclusion is indeed achieved in \cite {BBEGZ} by further using the interplay between the Mabuchi and Ding functionals. (Similar arguments appear in \cite {DR-iter} concerning the normalized Ricci iterations.)\end{rmk}In our Fano generalized Kähler setting, one can use the definition of the extended Mabuchi energy to also define an extension of the Ding functional as follows. For a generalized Kähler structure (g, I, J) of symplectic type with symplectic form F\in 2\pi c_1(M, J) and corresponding Kähler reduction \overline{\omega }, we denote by \mathring{h}_F the \emph {normalized} \rho ^B_I-potential, defined by\begin{equation*}\rho ^B_I(g) - \overline{\omega }= {\mathrmsl d}{\mathrmsl d}_J^c \mathring{h}_F, \qquad \int _M e^{2\mathring{h}_F}F^{[m]}=1,\end{equation*}and introduce the corresponding Ding functional (see \cite {Ding-Tian}) by the formula\begin{equation}{\bf D}(F) := {\bf M}(F) +2\left[\int _M \mathring{h}_{F}F^{[m]}- \int _M \mathring{h}_{0} \overline{\omega }_0^{[m]}\right],\end{equation}where \overline{\omega }_0\in 2\pi c_1(M, J) is a background Kähler metric with normalized Ricci potential \mathring{h}_0, i.e. \rho _0 - \overline{\omega }_0 = {\mathrmsl d}{\mathrmsl d}^c_J \mathring{h}_0, \int _M e^{2\mathring{h}_0} \overline{\omega }_0^{[m]}=1.Using Proposition~\ref {p:symplectic-Ricci-potential} and writing \overline{\omega }= \overline{\omega }_0 + {\mathrmsl d}{\mathrmsl d}^c_J \varphi , one can equivalently express {\bf D}(F) as\begin{equation}{\bf D}(F) = -2 I(\varphi ) - {V}\log \left(\int _M e^{-2\varphi }\left(\frac{\omega _{J}^{[m]}}{F^{[m]}}\right)\mu _0\right), \, \qquad \mu _0:= e^{2\mathring{h}_{0}}\overline{\omega }_0^{[m]}, \, \, V=\mathit {vol}(M, \overline{\omega }_0).\end{equation}When F=\omega _J=\overline{\omega } is a Kähler structure, the description (\ref {D-B}) reduces to the familiar expression of the Ding functional on a Fano Kähler manifold. Using Jensen^{\prime }s inequality in (\ref {Ding}) and the pointwise inequality F^{[m]} \ge \omega _J^{[m]} in (\ref {D-B}), we have the inequalities{\bf M}(F) \ge {\bf D}(F) + 2 \int _M \mathring{h}_{0} \overline{\omega }_0^{[m]} \ge {\bf D}(\overline{\omega }) + 2 \int _M \mathring{h}_{0} \overline{\omega }_0^{[m]}.Beside these nice properties, and unlike the extended Mabuchi functional {\bf M}, the behavior of {\bf D} along the normalized GKRF (\ref {unnormalized}) on a toric Fano manifold is as yet unclear.}}}\begin{thebibliography}{10}\end{thebibliography}\bibitem {Abreu1} M.~Abreu, {\it K{ä}hler geometry of toric varieties andextremal metrics}, Internat. J. Math {\bf 9} (1998), 641-651.}}\bibitem {Abreu2} M.~Abreu, {\it K{ä}hler metrics on toric orbifolds},J. Differential Geom. {\bf 58} (2001), 151-187.}\bibitem {apostolov-notes} V. 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